Iterative Scheme for Solving Optimal Transportation Problems Arising in Reflector Design

Table 1

Results of runs with the “Simple Discretization” Scheme 5 for several edge lengths ℎ. Memory was insufficient for the LP solver to handle the case ℎ=0.04, so the maximum number of mesh points that could be handled was about 1,750 on each aperture. The data used was the data from Section 4.1. Symbols used: ℎ: edge length; 𝑁: number of mesh points in 𝐷; 𝑀: number of mesh points in 𝑇; time (s): time in seconds; constr.: number of constraints.

ℎ	𝑀	𝑁	Max error 𝑅1	𝐿2 error 𝑅1	Max error 𝑅2	𝐿2 error 𝑅2	Time (s)	Constr.
0.09	511	522	3.68393𝐸−3	1.25443𝐸−3	4.58391𝐸−3	1.91131𝐸−3	2.97016	266,742
0.08	660	663	2.96216𝐸−3	7.30738𝐸−4	3.39579𝐸−3	1.08343𝐸−3	4.67627	437,580
0.07	880	874	2.33561𝐸−3	7.43916𝐸−4	2.83982𝐸−3	1.15916𝐸−3	10.1342	769,120
0.06	1,214	1,214	1.3051𝐸−3	7.12914𝐸−4	2.10579𝐸−3	8.28341𝐸−4	17.4949	1,473,796
0.05	1,752	1,757	9.11319𝐸−4	2.66152𝐸−4	1.20934𝐸−3	4.61313𝐸−4	45.9136	3,078,264